The Variability of the Slope Estimate

To construct a
confidence interval for the slope of the regression line,
we need to know the
standard error
of the
sampling distribution of the slope.
Many statistical software packages and some graphing calculators
provide the standard error of the slope as a regression analysis
output. The table below shows hypothetical output for the following
regression equation: y = 76 + 35x .

Predictor

Coef

SE Coef

T

P

Constant

76

30

2.53

0.01

X

35

20

1.75

0.04

In the output above, the standard error of the slope (shaded in gray)
is equal to 20. In this example, the standard error is referred to
as "SE Coeff". However, other software packages might use a
different label for the standard error. It might be "StDev",
"SE", "Std Dev", or something else.

If you need to calculate the standard error of the slope
(SE)
by hand, use the following formula:

SE = sb1 =
sqrt [ Σ(yi - ŷi)2
/ (n - 2) ]
/ sqrt [ Σ(xi -
x)2 ]

where yi is the value of the dependent variable for
observation i,
ŷi is estimated value of the dependent variable
for observation i,
xi is the observed value of the independent variable for
observation i,
x is the mean of the independent variable,
and n is the number of observations.

How to Find the Confidence Interval for the Slope of a
Regression Line

Specify the confidence interval. The range of the confidence
interval is defined by the sample statistic+margin of error. And the uncertainty is denoted
by the confidence level.

In the next section, we work through a problem that shows how to
use this approach to construct a confidence interval for the
slope of a regression line. Note that this approach is used for
simple linear regression (one independent variable and one dependent variable).

Test Your Understanding

Problem 1

The local utility company surveys 101 randomly selected
customers. For each survey participant, the company collects
the following: annual electric bill (in dollars) and home size
(in square feet). Output from a regression analysis
appears below.

Regression equation:
Annual bill = 0.55 * Home size + 15

Predictor

Coef

SE Coef

T

P

Constant

15

3

5.0

0.00

Home size

0.55

0.24

2.29

0.01

What is the 99% confidence interval for the slope of the regression
line?

The correct answer is (C). Use the following
four-step approach to construct a confidence interval.

Identify a sample statistic. Since we are trying to estimate
the slope of the true regression line, we use the
regression coefficient for home size (i.e., the sample estimate of
slope) as the sample statistic. From the regression output, we
see that the slope coefficient is 0.55.

Select a confidence level. In this analysis, the confidence level
is defined for us in the problem. We are working with a 99%
confidence level.

Find standard deviation or standard error. The standard
error is given in the regression output. It is 0.24.

Find critical value. The critical value is a factor used to
compute the margin of error. With simple linear regression,
to compute a confidence interval for the slope,
the critical value is a
t score
with
degrees of freedom equal to n - 2.
To find the critical value, we take these steps.

Specify the confidence interval. The range of the confidence
interval is defined by the sample statistic+margin of error. And the uncertainty is denoted
by the confidence level.

Therefore, the 99% confidence interval for this sample is 0.55 + 0.63, which is -0.08 to 1.18

If we replicated the same
study multiple times with different random samples and computed a confidence interval for each sample, we would expect
99% of the confidence intervals to contain the true slope of the regression line.